"Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at , if and only if it is equal to its Taylor's polynomialin some open neighborhood of, so showing it satisfies the Cauchy-Riemann equationat z= 0doesn't really prove it is analytic atanypoint. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?